# Euler phi at a product

If the positive greatest common divisor of the integers $a$ and $b$ is $d$, then

 $\varphi(ab)\;=\;\frac{\varphi(a)\,\varphi(b)\,d}{\varphi(d)}.$

Proof.  Using the positive prime factors $p$, the right hand side of the asserted equation is

 $\displaystyle\frac{d\cdot a\prod_{p\mid a}\frac{p-1}{p}\cdot b\prod_{p\mid b}% \frac{p-1}{p}}{d\prod_{p\mid a,\,p\mid b}\frac{p-1}{p}}$ $\displaystyle\;=\;\frac{ab\prod_{p\mid a,\,p\nmid b}\frac{p-1}{p}\cdot\prod_{p% \mid a,\,p\mid b}\frac{p-1}{p}\cdot\prod_{p\mid b,\,p\nmid a}\frac{p-1}{p}% \cdot\prod_{p\mid b,\,p\mid a}\frac{p-1}{p}}{\prod_{p\mid a,\,p\mid b}\frac{p-% 1}{p}}$ $\displaystyle\;=\;ab\prod_{p\mid a\;\lor\;p\mid b}\frac{p\!-\!1}{p}\;=\;ab% \prod_{p\mid ab}\frac{p\!-\!1}{p}\;=\;\varphi(ab),$

Q.E.D.

Title Euler phi at a product EulerPhiAtAProduct 2014-02-18 14:02:24 2014-02-18 14:02:24 pahio (2872) pahio (2872) 6 pahio (2872) Theorem msc 11A25 msc 11-00 EulerPhifunction DivisibilityByPrimeNumber